After months of sparring, Virginia moved closer Friday to exerting more influence over the board that operates Dulles and Reagan airports and supervises construction of the new Metrorail line to Loudoun County.

The Virginia House of Delegates unanimously passed a bill that would give Gov. Robert F. McDonnell (R) until July 1 to appoint two new people to the board of the Metropolitan Washington Airports Authority, which federal legislation recently increased in size from 13 to 17. The Senate previously passed the bill, which now heads to McDonnell for his signature.

McDonnell also appointed Todd A. Stottlemyer, chief executive of IT contractor Acentia, to replace longtime member Mame Reiley. Reiley, who had about two years left in her term, resigned last week because she is battling cancer.

Virginia has sought to increase its influence on the board, saying it lacks transparency in its actions. McDonnell had moved last year to appoint Stottlemyer and Caren Merrick, a businesswoman who ran unsuccessfully for the state Senate, after Congress increased the size of the board.

The board formerly consisted of five members appointed by Virginia, three each by the District and the federal government, and two by Maryland. The new structure gives Virginia two new members, and Maryland and the District gain one each.

A legal debate over whether the authority’s governing agreement needs to be changed has delayed McDonnell’s appointments. MWAA board members have said they think that the District must pass similar legislation before the board can be expanded.

MWAA Chairman Michael A. Curto welcomed Stottlemyer to the board Friday.

Mame was a dedicated and passionate voice on the Board and her many contributions will long be remembered,” he wrote in a statement. “Over the last couple of months, I have had the chance to get to know Todd, and I am greatly impressed with his professional accomplishments and his genuine interest in the operations of MWAA. I have every confidence he will serve with distinction.”

On Thursday, the Virginia House and Senate also passed identical bills that would prohibit the use of mandatory project labor agreements for state-funded projects in Virginia.

The move comes just as the airports authority is preparing to request bids for contractors to build the second phase of the $6 billion Silver Line to Dulles International Airport and Loudoun.

Phase 1, from Falls Church to Wiehle Avenue in Reston, is under construction and is scheduled for completion next year. There is a voluntary project labor agreement on the first phase.

Some legislators have pushed for the airports authority to nix the requirement for a project labor agreement because they think it would favor labor unions.

“We commend the legislature for firmly upholding Virginia’s long-standing right-to-work tradition,’’ said Jim Corcoran, president and chief executive of the Fairfax County Chamber of Commerce.

“The Northern Virginia business community adamantly opposes mandatory [project labor agreements] because they dissuade merit-based employers from participating in bidding processes, thereby hindering competition in the marketplace,” he said. “Open competition, free from constraints . . . ensures a level playing field for all contractors.”

MWAA officials would not comment on the labor agreement bills, which now head to the governor.

The airports authority earlier reversed a decision to make a project labor agreement mandatory for Phase 2 after Virginia politicians threatened to withhold $150 million McDonnell has promised to help pay for the project.

MWAA plans to give bidders a 10 percent bonus in scoring their proposals if they implement an agreement.